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There is no phase change between refracted (transmitted) rays and incident rays, according to boundary conditions of electric field and magnetic field, in accordance with the Maxwell's equations and the continuity of fields. But in Fourier optics a thin lens behaves as a phase transformation device, and in the derivation (from Goodman's "Introduction to Fourier optics") it is shown that the phase change is: $$p_L=e^\frac{-ik\,(x^2+y^2)}{2f}$$ where $k$ is the wavenumber, $f$ is the focal length. How do the two facts agree with each other?

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A lens has curved surfaces. The field is continuous across the curved interfaces between the lens and the surrounding medium (e.g. air). The equation you wrote is for the phase measured across a plane perpendicular to the axis of an optical system, and this plane does not coincide with any the surfaces of the lens.

In fact (under approximations that I do not have in mind in a rigorous way while I am writing, see below) one can obtain the "lens phase" you wrote by considering incident rays parallel to the optical axis and pretending that they are not bent by the lens.

One calculates the optical path distance (OPD) a ray travels through between two planes perpendicular to the optical axis, one placed before the lens, one placed after the lens. Keeping in mind that the OPD is equal to the distance travelled times the refractive index of the medium the ray is travelling through, you can probably see "with the eyes of your mind" that the OPD is larger where the lens is thicker.

Note that to do this calculation you assume that the phase is continuous, as for a small path the OPD is automatically small.

For what regards the approximation that allows one to consider the rays as propagating "straight through" the lens: the lens must be thin enough that the effect it has on the phase of the light field does not turn into an effect on the intensity of light when looking at the light field just across the lens. That is, consider a ray that is bent by the lens and consider its sideways displacement between "just before" and "just after" the lens: this displacement must be small.

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There is no phase change between refracted (transmitted) rays and incident rays, according to boundary conditions

That boundary condition applies only to the infinitesimal boundary between a medium (air) and the refracting material (glass); a thin lens formula, however, applies to a significant volume of material (the glass in the lens).

The 'thin lens' approximation is not intended to be thin compared to a wavelength of light, so it does not preclude a significant phase shift. If we take 'light' to be 500 nm wavelength (yellow), and index of refraction 1.5, a 1mm thick pane of glass might be 'thin' in the thin lens sense, but it accomplishes a phase shift of 6200 radians.

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  • $\begingroup$ The glass in the lens and the air which fill the remaining space between the two planes perpendicular to the optical axis which contain the lens. $\endgroup$ – JTS Apr 13 at 19:31

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